Computational Mechanics (2019) 63:1315–1331https://doi.org/10.1007/s00466-018-1651-0

ORIG INAL PAPER

Aorta zero-stress state modeling with T-spline discretization

Takafumi Sasaki1 · Kenji Takizawa1 · Tayfun E. Tezduyar2,3

Received: 4 September 2018 / Accepted: 26 October 2018 / Published online: 2 November 2018© The Author(s) 2018

AbstractThe image-based arterial geometries used in patient-specific arterial fluid–structure interaction (FSI) computations, such asaorta FSI computations, do not come from the zero-stress state (ZSS) of the artery. We propose a method for estimating theZSS required in the computations. Our estimate is based on T-spline discretization of the arterial wall and is in the form ofintegration-point-based ZSS (IPBZSS). The method has two main components. (1) An iterative method, which starts with acalculated initial guess, is used for computing the IPBZSS such that when a given pressure load is applied, the image-basedtarget shape is matched. (2) A method, which is based on the shell model of the artery, is used for calculating the initialguess. The T-spline discretization enables dealing with complex arterial geometries, such as an aorta model with branches,while retaining the desirable features of isogeometric discretization. With higher-order basis functions of the isogeometricdiscretization, we may be able to achieve a similar level of accuracy as with the linear basis functions, but using larger-sizeand much fewer elements. In addition, the higher-order basis functions allow representation of more complex shapes withinan element. The IPBZSS is a convenient representation of the ZSS because with isogeometric discretization, especially withT-spline discretization, specifying conditions at integration points is more straightforward than imposing conditions on controlpoints. Calculating the initial guess based on the shell model of the artery results in a more realistic initial guess. To showhow the new ZSS estimation method performs, we first present 3D test computations with a Y-shaped tube. Then we show a3D computation where the target geometry is coming from medical image of a human aorta, and we include the branches inour model.

Keywords Patient-specific arterial FSI · Image-based geometry · Aorta · Zero-stress state · Isogeometric wall discretization ·T-spline basis functions · Integration-point-based zero-stress state · Shell-model-based initial guess

1 Introduction

The patient-specific arterial fluid–structure interaction (FSI)computations reported in [1–4] were among the earliest ofits kind. The core method in these computations was theearly version of the Deforming-Spatial-Domain/StabilizedSpace–Time (DSD/SST) method [5,6], which is now called

B Kenji TakizawaKenji.Takizawa@tafsm.org

Tayfun E. Tezduyartezduyar@tafsm.org

1 Department of Modern Mechanical Engineering, WasedaUniversity, 3-4-1 Ookubo, Shinjuku-ku, Tokyo 169-8555,Japan

2 Mechanical Engineering, Rice University, MS 321,6100 Main Street, Houston, TX 77005, USA

3 Faculty of Science and Engineering, Waseda University,3-4-1 Ookubo, Shinjuku-ku, Tokyo 169-8555, Japan

the “ST-SUPS.”The acronym“SUPS” indicates the stabiliza-tion components, the Streamline-Upwind/Petrov–Galerkin(SUPG) [7] and Pressure-Stabilizing/Petrov–Galerkin(PSPG) [5] stabilizations.

The ST computations have been only a small part ofthe large number of cardiovascular fluid mechanics and FSIcomputations seen in the last 15years (see, for example, [8–29]), with the Arbitrary Lagrangian–Eulerian (ALE) method[30] having the largest share in the computations reported.Still, a large number of computations with the ST methodswere also reported in the last 15years. In the first 8yearsof that period the ST computations were for FSI of abdom-inal aorta [31], carotid artery [31] and cerebral aneurysms[32–38]. In the last 7years, the ST computations focusedon even more challenging aspects of cardiovascular fluidmechanics andFSI, including comparative studies of cerebralaneurysms [39,40], stent treatment of cerebral aneurysms[41–45], heart valve flow computation [46–51], aorta flow

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analysis [51–54], and coronary arterial dynamics [55]. Thelarge number of computational challenges encountered wereaddressed by the advances in core methods for movingboundaries and interfaces (MBI) and FSI (see, for exam-ple, [20,21,40,46,47,49,50,56–64] and references therein)and in special methods targeting cardiovascularMBI and FSI(see, for example, [20,38,44,45,48–51,54,65] and referencestherein).

A challenge very specific to patient-specific arterial FSIcomputations, such as patient-specific aorta FSI computa-tions, is how to use the image-based arterial geometry. Theimage-based geometry does not come from the zero-stressstate (ZSS) of the artery. Special methods targeting cardio-vascular MBI and FSI include those designed to account forthat. The attempt to find a ZSS for the artery in the FSI com-putation was first made in a 2007 conference paper [66],where the concept of estimated zero-pressure (EZP) arterialgeometry was introduced. The method introduced in [66] forcalculating an EZP geometry was also included in a 2008journal paper on ST arterial FSI methods [32], as “a rudi-mentary technique” for addressing the issue. It was pointedout in [32,66] that quite often the image-based geometrieswere used as arterial geometries corresponding to zero bloodpressure, and that it would be more realistic to use the image-based geometry as the arterial geometry corresponding tothe time-averaged value of the blood pressure. Given thearterial geometry at the time-averaged pressure value, an esti-mated arterial geometry corresponding to zero bloodpressureneeded to be built. Special methods developed to address theissue include the newer EZP versions [20,34,37,38,65] andthe prestress technique introduced in [16], which was refinedin[18] and presented also in [20,65].

We introduced in [67] a method for estimation of theelement-based ZSS (EBZSS) in the context of finite ele-ment discretization of the arterial wall. The method hasthreemain components. (1) An iterativemethod, which startswith a calculated initial guess, is used for computing theEBZSS such that when a given pressure load is applied,the image-based target shape is matched. (2) A method forstraight-tube segments is used for computing the EBZSS sothat we match the given diameter and longitudinal stretchin the target configuration and the “opening angle.” (3)An element-based mapping between the artery and straight-tube is extracted from the mapping between the artery andstraight-tube segments. This provides the mapping from thearterial configuration to the straight-tube configuration, andfrom the estimated EBZSS of the straight-tube configurationback to the arterial configuration, to be used as the initialguess for the iterative method that matches the image-basedtarget shape. Test computations with the method were alsopresented in [67] for straight-tube configurations with singleand three layers, and for a curved-tube configuration withsingle layer. The method was used also in [55] in coronary

arterial dynamics computations with medical-image-basedtime-dependent anatomical models.

In [68], we introduced the version of the EBZSS esti-mation method with isogeometric wall discretization, usingNURBS basis functions. With isogeometric discretization,we can obtain the element-based mapping directly, insteadof extracting it from the mapping between the artery andstraight-tube segments. That is because all we need for theelement-based mapping, including the curvatures, can beobtained within an element. With NURBS basis functions,we may be able to achieve a similar level of accuracy as withthe linear basis functions, but using larger-size and muchfewer elements, and the NURBS basis functions allow repre-sentation of more complex shapes within an element. The 2Dtest computationswith straight-tube configurations presentedin [68] showed how the EBZSS estimation method withNURBS discretization works. In [69], which is an expanded,journal version of [68], we also showed how the method canbe used in a 3D computation where the target geometry iscoming from medical image of a human aorta.

In this articlewe are introducing a newmethod for estimat-ing the ZSS. The estimate is based on T-spline discretizationof the arterial wall and is in the form of integration-point-basedZSS (IPBZSS). Themethod has twomain components.(1) An iterative method, which starts with a calculated initialguess, is used for computing the IPBZSS such that when agiven pressure load is applied, the image-based target shapeis matched. (2) A method, which is based on the shell modelof the artery, is used for calculating the initial guess. TheT-spline discretization enables dealing with complex arte-rial geometries, such as an aorta model with branches, whileretaining the desirable features of isogeometric discretiza-tion. The IPBZSS is a convenient representation of the ZSSbecause with isogeometric discretization, especially withT-spline discretization, specifying conditions at integrationpoints is more straightforward than imposing conditions oncontrol points. Calculating the initial guess based on the shellmodel of the artery results in a more realistic initial guess. Toshow how the new method for estimating the ZSS performs,we first present 3D test computations with a Y-shaped tube.Then we show a 3D computation where the target geome-try is coming from medical image of a human aorta, and weinclude the branches in our model.

In Sect. 2, we describe the Element-Based TotalLagrangian (EBTL) method, including the EBZSS andIPBZSS concepts. How the initial guess is calculated basedon the shell model of the artery is described in Sect. 3. Thenumerical examples are given in Sect. 4, and the concludingremarks in Sect. 5.

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2 EBTLmethod

In this section we provide an overview of the EBTL method[67], including the EBZSS concept, and describe the IPBZSSconcept and the conversion between the two ZSS.

Let Ω0 ⊂ Rnsd be the material domain of a structure in

the ZSS, where nsd is the number of space dimensions, andlet Γ0 be its boundary. Let Ωt ⊂ R

nsd , t ∈ (0, T ), be thematerial domain of the structure in the deformed state, andlet Γt be its boundary. The structural mechanics equationsbased on the total Lagrangian formulation can be written as

∫Ω0

w · ρ0d2ydt2

dΩ +∫

Ω0

δE : S dΩ −∫

Ω0

w · ρ0f dΩ

=∫

(Γt )h

w · h dΓ . (1)

Here, y is the displacement, w is the virtual displacement,δE is the variation of the Green–Lagrange strain tensor, Sis the second Piola–Kirchhoff stress tensor, ρ0 is the massdensity in the ZSS, f is the body force per unit mass, and h isthe external stress vector applied on the subset (Γt )h of theboundary Γt .

2.1 EBZSS

In the EBTLmethod the ZSS is definedwith a set of positionsXe0 for each element e. Positions of nodes from different

elements mapping to the same node in the mesh do not haveto be the same. In the reference state, XREF, all elementsare connected by nodes, and we measure the displacement yfrom that connected state. The implementation of themethodis simple. The deformation gradient tensor F is evaluated foreach element:

Fe ≡ ∂x∂Xe

0, (2)

= ∂ (XREF + y)∂Xe

0, (3)

where x is the position in the deformed configuration. Thedeformation gradient tensors for different elements are ondifferent states, but the terms in Eq. (1), including the secondterm, do not depend on the orientation. Therefore the restof the process is the same as it is in the total Lagrangianformulation.

2.2 IPBZSS

The key idea behind the EBZSS method was that, due to theobjectivity, all the quantities seen in Eq. (1) can be computedwith any orientation of the ZSS. We can extend the way wesee Xe

0 to integration-point counterpart of Xe0. As we did

with the EBZSS, we work with the reference domain. Withthe reference Jacobian

JREF = det

(∂XREF

∂X0

), (4)

Eq. (1) can be rearranged as

∫ΩREF

w · ρ0d2ydt2

J−1REFdΩ +

∫ΩREF

δE : S J−1REFdΩ

−∫

ΩREF

w · ρ0f J−1REFdΩ =

∫(Γt )h

w · h dΓ . (5)

In our implementation, we use the natural coordinates, withcovariant basis vectors

gI = ∂x∂ξ I

, (6)

GI = ∂X∂ξ I

, (7)

where ξ I is the parametric coordinate, and I = 1, . . . , npd,with npd being the number of parametric dimensions. Thecontravariant basis vectors can be calculated with the metrictensor components as

gI = gI JgJ , (8)

GI = GI JGJ , (9)

where

gI J = gI · gJ , (10)

GI J = GI · GJ , (11)[gI J

]= [gI J ]

−1 , (12)[GI J

]= [GI J ]

−1 . (13)

With those vectors, we can express the deformation gradienttensor:

F = gIGI , (14)

and the Cauchy–Green deformation tensor:

C = FT · F (15)

= GIgI · gJGJ (16)

= gI JGIGJ . (17)

The Jacobian, J = det F, can be expressed as

J 2 = detC. (18)

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1318 Computational Mechanics (2019) 63:1315–1331

We can write detC as

detC = det [gI J ]

det [GI J ], (19)

and from that we obtain

J =(det [gI J ]

det [GI J ]

) 12

. (20)

We define the covariant basis vectors corresponding toXREF:

(GREF)I = ∂XREF

∂ξ I, (21)

and the components of the metric tensor are

(GREF)I J = (GREF)I · (GREF)J . (22)

In Eq. (20), we replace gI and gJ with (GREF)I and (GREF)Jand obtain an alternative to the expression given by Eq. (4):

JREF =(det

[(GREF)I J

]det [GI J ]

) 12

. (23)

The Green–Lagrange strain tensor,

E = 1

2(C − I) , (24)

where I is the identity tensor, can be expressed with the con-travariant basis vectors as

E = 1

2(gI J − GI J )GIGJ . (25)

The second Piola–Kirchhoff tensor can be expressedwith thecovariant basis vectors as

S = SI JGIGJ , (26)

where SI J can be expressed with the components of themetric tensors. Thus, the inner product δE : S, and all theother quantities, in the weak form given by Eq. (5) can beevaluated without actually using the basis vectors GI . Thisjustifies using (GI J )k as the integration-point counterpart ofXe0, with k = 1, . . . , nint, where nint is the number of inte-

gration points. Note thatGI J is symmetric, and therefore theIPBZSS representationwill in 3Dhave 6×nint parameters foreach element.

2.3 EBZSS to IPBZSS

Converting the EBZSS representation to IPBZSS represen-tation is straightforward. From givenXe

0 we can calculate thecovariant basis vectors at each integration point ξξξ k :

(GI )k = ∂Xe0

∂ξ I

∣∣∣∣ξξξ=ξξξ k

, (27)

and obtain the components of themetric tensor fromEq. (11).

2.4 IPBZSS to EBZSS

Converting the IPBZSS representation to EBZSS represen-tation, which we might need for visualization purposes, will,in general, not be exact because the IPBZSS hasmore param-eters than the EBZSS. Given (GI J )k , we solve a steady-stateelement-based problem (with f = 0 and h = 0):

∫Ωe

REF

δE : S J−1REFdΩ = 0, (28)

and the solution to that, in the form XREF + y, will be theEBZSS representation. If the stress calculated from the solu-tion is zero, then the conversion will be exact. We note thatto obtain a steady-state solution, we need to preclude trans-lation and rigid-body rotation by imposing 6 appropriateconstraints. To do that we first select three control points:A, B and C . We set all three components of yA to be zeroand constrain yB to be in the direction (XREF)B − (XREF)A.The last constraint is yC to be on the plane defined by thevector ((XREF)B − (XREF)A) × ((XREF)C − (XREF)A).

2.5 An iterative method

Here we assume that we have a reasonably good initial guessfor the IPBZSS (see Sect. 3). The iterative method below isused in calculating the IPBZSS that results in the target stateassociated with the given load. In the iterative method, weestimate F from the i th solution. We simply assume

Fi+1 = Fi , (29)

which means

(Fi+1

)−1 =(Fi)−1

. (30)

The inverse of the deformation gradient tensor from the i thsolution can be written as

(Fi)−1 = (GI )

i(gI)i

. (31)

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Computational Mechanics (2019) 63:1315–1331 1319

Similarly, the inverse of the target deformation gradient ten-sor is

(Fi+1

)−1 = (GI )i+1 (GREF)

I . (32)

Thus, we obtain the following equation:

(GK )i+1 (GREF)K = (GK )i

(gK

)i. (33)

Inner-producting both sides of this equation from the rightwith the covariant basis vectors corresponding to XREF, weobtain

(GK )i+1 (GREF)K · (GREF)I = (GK )i

(gK

)i · (GREF)I ,

(34)

which results in

(GI )i+1 = (GK )i

(gK

)i · (GREF)I . (35)

The components of the metric tensor are

(GI J )i+1 =

((GK )i

(gK

)i · (GREF)I

)

·(

(GL)i(gL)i · (GREF)J

). (36)

Rearranging the terms, we obtain

(GI J )i+1 = (GKL)i

((gK

)i · (GREF)I

)

×((

gL)i · (GREF)J

). (37)

Thus, we update the components of the metric tensor withoutactually knowing the basis vectors GI .

Once we obtain the IPBZSS at the end of the iterations,we will have the option to convert it to EBZSS and use thatin the subsequent y computations.

3 Initial guess based on the shell model ofthe artery

An analytical relationship between the ZS and referencestates of straight-tube segments was given in [67]. The rela-tionship was called “straight-tube ZSS template” in [69]and was extended to curved tubes. These were for theEBZSS. Here we directly build the IPBZSS instead ofbuilding the EBZSS first. This is simpler because withisogeometric discretization, especially with T-spline dis-cretization, specifying conditions at integration points is far

more straightforward than imposing conditions on controlpoints.

We start with the artery inner surface, which is what themedical images show. Typically, we cannot discern the wallthickness from the medical image. Therefore we first buildthe inner-surface mesh with T-splines. Then we build a T-spline volume mesh by extruding the surface elements by anestimated thickness.

In our notation here, x will now implyXREF, which is our“target” shape, and X will imply X0. We explain the methodin the context of one element in the thickness direction.Extending the method to multiple elements is straightfor-ward.

3.1 Inner-surface coordinates in the target state

The coordinate systemwe have here is similar to the one usedfor the shell modeling in [70]. We note that the “midsurface”of the shell formulation has been shifted to the inner surfacehere, and • indicates the inner surface. The basis vectors are

gα = ∂x∂ξα

, (38)

where α = 1, . . . , nsd − 1, and the third direction is

n = g1 × g2∥∥g1 × g2∥∥ . (39)

The second fundamental form is defined as

bαβ = ∂gα

∂ξβ· n, (40)

and the curvature tensor is

κ̂κκ = −bαβ︸ ︷︷ ︸κ̂αβ

gαgβ. (41)

Clearly, κ̂κκ is symmetric.For a given unit vector t on the surface, we obtain the

curvature κ̂ as

κ̂ = t · κ̂κκ · t. (42)

If t is a principal direction,

κ̂κκ · t = κ̂t, (43)

and κ̂ is the corresponding principal curvature. The eigen-vector can be expressed as

t = tβgβ. (44)

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Substituting this into Eq. (43) and inner-producting with gα ,we obtain

gγα

(κ̂α•β − κ̂δα

β

)tβ = 0, (45)

where the mixed components indicate

κ̂κκ = κ̂α•βgαgβ. (46)

Since the inverse of[gγα

]exists,

det[κ̂α•β − κ̂δα

β

]= 0, (47)

κ̂ is an eigenvalue of the matrix defined by the mixed com-ponents κ̂α•β , and we will call the two eigenvalues κ̂1 and κ̂2.For κ̂1 > κ̂2 we obtain the corresponding eigenvectors:

t1 = (t1)β gβ, (48)

t2 = (t2)β gβ, (49)

where t1 and t2 are unit vectors. From Eq. (43), we write

κ̂κκ · t1 = κ̂1t1, (50)

κ̂κκ · t2 = κ̂2t2. (51)

Since κ̂κκ is symmetric,

t2 · κ̂κκ · t1 = t1 · κ̂κκ · t2. (52)

Substituting Eqs. (50) and (51) into this, we obtain

κ̂1t1 · t2 = κ̂2t1 · t2. (53)

Thus, the two vectors are orthonormal, and we can expressthe curvature tensor as

κ̂κκ = κ̂1t1t1 + κ̂2t2t2. (54)

When κ̂1 = κ̂2, an arbitrary orthonormal set of t1 and t2 canbe used in the above equation.

For more details on calculating the eigenvalues and eigen-vectors, see Appendix A.

3.2 Inner-surface coordinates in the ZSS

Since the principal curvature directions t1 and t2 of the targetshape are orthogonal to each other, we can build the ZSSshape using those directions. The basis vectors on the innersurface in the ZSS are

Gα = ∂X∂ξα

, (55)

and the third direction is

N = G1 × G2∥∥G1 × G2∥∥ . (56)

The stretches corresponding to those directions will be λ̂1and λ̂2. Then the ZSS basis vectors are calculated from

λ̂1t1 = F · t1, (57)

λ̂2t2 = F · t2. (58)

That is

λ̂1F−1 · t1 = t1, (59)

λ̂2F−1 · t2 = t2. (60)

Because the third direction is orthogonal to t1 and t2, we canreduce these equations to

λ̂1Gβgβ · t1 = t1, (61)

λ̂2Gβgβ · t2 = t2. (62)

Substituting Eqs. (48) and (49) into these, we get

λ̂1 (t1)α Gα = (t1)

α gα, (63)

λ̂2 (t2)α Gα = (t2)

α gα. (64)

This can also be written as

[λ̂1 0

0 λ̂2

][(t1)1 (t1)2

(t2)1 (t2)2

][G1

G2

]=[(t1)1 (t1)2

(t2)1 (t2)2

][g1g2

],

(65)

and from that we calculate the basis vectors as

[G1

G2

]=[(t1)

1 (t1)2

(t2)1 (t2)

2

]−1⎡⎣

1λ̂1

0

0 1λ̂2

⎤⎦[(t1)

1 (t1)2

(t2)1 (t2)

2

][g1

g2

].

(66)

3.3 Wall coordinates in the target state

The position in the target configuration is

x = x + nϑ, (67)

where 0 ≤ ϑ ≤ hth, and hth is the wall thickness in the targetconfiguration. The basis vectors will vary along the thicknessdirection as

gα = ∂x∂ξα

(68)

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Computational Mechanics (2019) 63:1315–1331 1321

= gα + ∂n∂ξα

ϑ (69)

= gα − bαγ gγ ϑ. (70)

The second and third lines are explained in Appendix 1. Thethird coordinate is mapped as

ϑ = 1 + ξ3

2hth, (71)

where −1 ≤ ξ3 ≤ 1. The basis vector in the third directionis constant as

g3 = hth2n, (72)

and

g3 = 2

hthn. (73)

With that, the components of the metric tensor are

gαβ = gαβ − 2bαβϑ + bαγ gγ δbβδϑ

2, (74)

g3α = 0, (75)

gα3 = 0, (76)

g33 = h2th4

. (77)

3.4 Wall coordinates in the ZSS

The position in the ZSS configuration is

X = X + Nϑ0, (78)

where 0 ≤ ϑ0 ≤ (hth)0, and (hth)0 is the wall thickness inthe ZSS configuration. The basis vectors will vary along thethickness direction as

Gα = ∂X∂ξα

(79)

= Gα + ∂N∂ξα

ϑ0 (80)

= Gα − BαγGγϑ0. (81)

The curvature tensor in the ZSS configuration is

κ̂κκ0 = (κ̂0)1 t1t1 + (

κ̂0)2 t2t2. (82)

From that,

Bαβ = −κ̂κκ0 : GαGβ (83)

= − (κ̂0)1

(t1 · Gα

) (t1 · Gβ

)

− (κ̂0)2

(t2 · Gα

) (t2 · Gβ

). (84)

Similar to what we had for t1 and t2,

λ3F−1 · n = n, (85)

which becomes

λ3G3g3 · n = n. (86)

We substitute Eq. (73) into this and obtain

G3 = hth2λ3

n, (87)

and

G3 = 2λ3hth

n. (88)

3.5 Calculating the components of the ZSSmetrictensor at each integration point

For an integration point ξξξ , we can obtain the components ofthe metric tensor as

Gαβ = Gαβ − 2Bαβϑ0 + Bαγ Gγ δBβδϑ

20 , (89)

G3α = 0, (90)

Gα3 = 0, (91)

G33 = h2th4λ23

. (92)

The third coordinate can be obtained from

ϑ0 =∫ ξ3

−1

hth2λ3

dξ3. (93)

Assuming incompressible material, J = 1,

λ3 = A0

A, (94)

where

A2 = det[gαβ

], (95)

A20 = det

[Gαβ

]. (96)

The components of the matrix tensors are given by Eqs. (74)and (89).

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1322 Computational Mechanics (2019) 63:1315–1331

Fig. 1 Y-shaped tube. Target state. The end diameters are 20, 14 and10mm

Fig. 2 Y-shaped tube. Mesh made of cubic and quartic T-splines. Redcircles represent the control points. The parts with the quartic T-splines,obtained by order elevation [72], are around the two extraordinarypoints, each connected to six edges. (Color figure online)

3.6 Design of the ZSS

The design parameters are the principal curvatures(κ̂0)1 and(

κ̂0)2, and the stretches λ̂1 and λ̂2 for each principal curvature

direction. Those parameters can be determined from κ̂1 andκ̂2 of the target configuration.

As proposed in [69], the two principal directions are seenas circumferential and longitudinal directions, and κ̂1 is inthe circumferential direction, giving us

(κ̂0)1 = 2π − φ

2πκ̂1. (97)

Here φ is the opening angle, which is seen after a longitudi-nal cut, based on artery experimental data [71]. The stretch inthat direction, λ̂1, corresponds to λθ in [69] and that is deter-

1.0 1.5 2.0

Thickness (mm)

Fig. 3 Y-shaped tube. Wall thickness distribution

Fig. 4 Y-shaped tube. The IPBZSS initial guess, shown using theEBZSS representation

mined from the 2D computations in [69]. We assume thatin the longitudinal direction the ZSS configuration has zerocurvature,

(κ̂0)2 = 0. The stretch in that direction, λ̂2, cor-

responds to λz in [69]. If at an integration point κ̂2 ≈ κ̂1, weset

(κ̂0)2 = (

κ̂0)1 and λ̂2 = λ̂1. That makes the assignment

of the principal directions less consequential.

4 Numerical examples

In the numerical examples we use the Fung’s model [20,38,70] with D1 = 2.6447×103 Pa, D2 = 8.365, and thePoisson’s ratio ν = 0.45. The energy-density function weuse is in the form

ϕFR (C) = D1

⎛⎝e

D2

(trJ− 2

3 C−3

)− 1

⎞⎠

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Computational Mechanics (2019) 63:1315–1331 1323

Fig. 5 Y-shaped tube. An element in the target state (top) and the corre-sponding IPBZSS initial guess, shown using the EBZSS representation(bottom)

10−6 10−4 10−2 100

Displacement (mm)

Fig. 6 Y-shaped tube. Colored by ‖y‖ computed from the convergedIPBZSS. (Color figure online)

+ 1

2κ

(1

2

(J 2 − 1

)− ln J

), (98)

where

κ = 4D1D2 (1 + ν)

3 (1 − 2ν). (99)

0.1 0.2 0.5 1.0

Mises strain

Fig. 7 Y-shaped tube. The von Mises strain, from the IPBZSS initialguess (top) and from the converged IPBZSS (bottom)

Fig. 8 Y-shaped tube. The converged IPBZSS element correspondingto the element in Fig. 5, shown using the EBZSS representation

We assume the pressure associated with the target shape is92 mm Hg.

The initial guess for the iterations is determined asdescribed in Sect. 3, with φ = 5

2π and λ̂2 = 1.05. After theiterations, explained in Sect. 2.5, for comparison purposes,we convert the IPBZSS representation to EBZSS representa-tion,with themethod described in Sect. 2.4.With the EBZSS,

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1324 Computational Mechanics (2019) 63:1315–1331

10−6 10−4 10−2 100

Displacement (mm)

Fig. 9 Y-shaped tube. Colored by ‖y‖ computed after converting theconverged IPBZSS to EBZSS. (Color figure online)

0.1 0.2 0.5 1.0

Mises strain

Fig. 10 Y-shaped tube. The vonMises strain, obtained after convertingthe converged IPBZSS to EBZSS

we compute y again, and compare that to what we obtaineddirectly from the IPBZSS.

4.1 Y-shaped tube

The target state of the Y-shaped tube is shown in Fig. 1. Theend diameters of the tube are 20, 14 and 10mm. Figure2shows the T-spline mesh. The mesh is based on a mixture ofcubic and quartic T-splines. Thewall thickness distribution issmooth, outcome of solving the Laplace’s equation over theinner surface, with Dirichlet boundary conditions at the tubeends, where the value specified is 0.1 times the end diameter.Figure 3 shows the thickness distribution. The volume meshis built with one element (cubic Bézier element) in the thick-ness direction. The number of control points and elementsare 5,180 and 2,592.

Fig. 11 Patient-specific aortageometry. Target state, extractedfrom medical images

Fig. 12 Patient-specific aortageometry. Mesh made of cubicand quartic T-splines. Redcircles represent the controlpoints. The parts with thequartic T-splines, obtained byorder elevation [72], are aroundthe eight extraordinary points.(Color figure online)

Since the IPBZSS cannot be visualized, we show it usingthe EBZSS representation. Figure 4 shows the initial guessfor the IPBZSS. Figure 5 shows an element in the target stateand the corresponding IPBZSS initial guess.

We iterate and obtain the converged IPBZSS. Figure 6shows ‖y‖ computed from that. The maximum value of ‖y‖is 1.279×10−14 mm. Figure7 shows the von Mises strain,computed from the IPBZSS initial guess and from the con-verged IPBZSS. The converged IPBZSS element is shown inFig. 8.

Remark 1 Figure 8 shows that the opening angle and the lon-gitudinal stretch do not change much between the IPBZSSinitial guess and the converted IPBZSS. However, the cir-cumferential stretches are somewhat different. The initial

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Computational Mechanics (2019) 63:1315–1331 1325

0.0 1.0 2.0

Thickness (mm)

Fig. 13 Patient-specific aorta geometry. Wall thickness distribution

guess for the circumferential stretch was based on the 2Dcomputations reported in [69]. Alternatively, we can esti-mate the stretch by an analytical solution, similar to the onedescribed in [70].

We also compute y after converting the converged IPBZSSto EBZSS. Figure 9 shows ‖y‖ computed that way. The max-imum value of ‖y‖ is 1.626×10−2 mm. Figure 10 shows thevon Mises strain. There is no visible difference between thestrains obtained from the IPBZSS directly and after conver-sion to EBZSS.

4.2 Patient-specific aorta geometry

The target state of the patient-specific geometry is shown inFig. 11. Figure 12 shows the T-spline mesh.

The wall thickness distribution is smooth, outcome ofsolving the Laplace’s equation over the inner surface, withDirichlet boundary conditions at the tube ends, where thevalue specified is 0.08 times the end diameter. In some partsof the branched area the thickness exceeds the radius of cur-vature, and there we reduce the thickness to 0.8 times theradius of curvature. Figure 13 shows the thickness distribu-tion.

The volume mesh is built again with one element (cubicBézier element) in the thickness direction. The number ofcontrol points and elements are 9,244 and 4,360.

Figure 14 shows the initial guess for the IPBZSS. Weagain iterate and obtain the converged IPBZSS. Figure 15

Fig. 14 Patient-specific aortageometry. The IPBZSS initialguess, shown using the EBZSSrepresentation

10−6 10−4 10−2 100

Displacement (mm)

Fig. 15 Patient-specific aorta geometry.Colored by‖y‖ computed fromthe converged IPBZSS. (Color figure online)

shows ‖y‖ computed from that. The maximum value of ‖y‖is 1.163×10−13 mm. Figure16 shows the von Mises strain,computed from the IPBZSS initial guess and from the con-verged IPBZSS.

We again compute y after converting the convergedIPBZSS to EBZSS. Figure17 shows ‖y‖ computed that way.The maximum value of ‖y‖ is 1.057×10−1 mm. Figure 18shows the von Mises strain. These results indicate that theEBZSS obtained by conversion from the IPBZSS is also areasonable representation of the aorta ZSS.

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1326 Computational Mechanics (2019) 63:1315–1331

0.01 0.20 4.00

Mises strain

Fig. 16 Patient-specific aorta geometry. The vonMises strain, from theIPBZSS initial guess (top) and from the converged IPBZSS (bottom)

5 Concluding remarks

We have introduced a new method for estimating the ZSSrequired in patient-specific arterial FSI computations, wherethe image-based arterial geometries do not come from theZSS of the artery. The estimate is based on T-spline dis-cretization of the arterial wall and is in the form of IPBZSS.The method has two main components. (1) An iterativemethod, which starts with a calculated initial guess, is usedfor computing the IPBZSS such that when a given pressure

10−6 10−4 10−2 100

Displacement (mm)

Fig. 17 Patient-specific aorta geometry. Colored by ‖y‖ computed afterconverting the converged IPBZSS to EBZSS. (Color figure online)

0.01 0.20 4.00

Mises strain

Fig. 18 Patient-specific aorta geometry. The vonMises strain, obtainedafter converting the converged IPBZSS to EBZSS

load is applied, the image-based target shape is matched.(2) A method, which is based on the shell model of theartery, is used for calculating the initial guess. The T-splinediscretization enables dealing with complex arterial geome-tries, such as an aorta model with branches, while retaining

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the desirable features of isogeometric discretization. Thedesirable features of higher-order basis functions of isoge-ometric discretization include being able to achieve a similarlevel of accuracy as with the linear basis functions, butusing larger-size and much fewer elements, and being ableto represent more complex shapes within an element. TheIPBZSS is a convenient representation of the ZSS becausewith isogeometric discretization, especially with T-splinediscretization, specifying conditions at integration points ismore straightforward than imposing conditions on controlpoints. Calculating the initial guess based on the shell modelof the artery results in a more realistic initial guess. To showhow the new ZSS estimation method performs, we first pre-sented 3D test computations with a Y-shaped tube. Then wepresented a 3D computation where the target geometry wascoming frommedical image of a human aorta, and the modelincluded the aorta branches.

Acknowledgements This work was supported in part by JST-CREST;Grant-in-Aid for Scientific Research (S) 26220002 from theMinistry ofEducation, Culture, Sports, Science and Technology of Japan (MEXT);Grant-in-Aid for Scientific Research (A) 18H04100 from Japan Soci-ety for the Promotion of Science; andRice–Waseda research agreement.This work was also supported (first author) in part by Grant-in-Aid forJSPS Research Fellow 18J14680. The mathematical model and com-putational method parts of the work were also supported (third author)in part by ARO Grant W911NF-17-1-0046 and Top Global UniversityProject of Waseda University.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

A Eigenvalues and eigenvectors of the curva-ture tensor

In calculating the eigenvalues from the quadratic equation

κ̂2 − κ̂tr[κ̂α•β

]+ det

[κ̂α•β

]= 0, (100)

we obtain

κ̂ =tr[κ̂α•β

]±√tr2

[κ̂α•β

]− 4 det

[κ̂α•β

]

2. (101)

This can also be written as

κ̂l =tr[κ̂α•β

]− (−1)l

√tr2

[κ̂α•β

]− 4 det

[κ̂α•β

]

2, (102)

with l = 1, 2. In calculating the eigenvectors, we work with(v1)

α and (v2)α , the contravariant components of the non-

normalized eigenvectors. The vectors are normalized withthe scaling factors τ1 and τ2:

(tl)α = τl (vl)

α . (103)

With that, we obtain the relationship

(τl)2 (vl)

α gαβ (vl)β = 1, (104)

which yields

τl = ((vl)

α gαβ (vl)β)− 1

2 . (105)

Given (v1)α , which we will explain later how to calculate,

from Eqs. (103) and (105) we obtain

(t1)α = (

(v1)γ gγ δ (v1)

δ)− 1

2 (v1)α . (106)

We calculate t2 from the orthogonality between t1 and t2:

t1 · t2 = 0. (107)

With the substitutions t1 = (t1)α gα and t2 = (t2)β gβ , weobtain

(t1)α (t2)α = 0. (108)

The expression

(v2)α = Rαβ (t1)β , (109)

where

[Rαβ

] =[0 1

−1 0

], (110)

satisfiesEq. (108). To calculate τ2,we start from the covariantversion of (105):

τ2 = ((v2)α gαβ (v2)β

)− 12 . (111)

Substituting Eq. (109) into Eq. (111), we obtain

(τ2)−2 = Rαγ (t1)

γ gαβ Rβδ (t1)δ . (112)

By expanding the matrices and rearranging, we obtain therelationship

Rαγ gαβ Rβδ = det[gγ δ] [gγ δ

]−1(113)

= det−1[gγ δ]gγ δ. (114)

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1328 Computational Mechanics (2019) 63:1315–1331

Combining Eqs. (112) and (114), we obtain

(τ2)−2 = det−1[gγ δ] gγ δ (t1)

γ (t1)δ

︸ ︷︷ ︸=1

, (115)

and then

τ2 = det12 [gγ δ]. (116)

Using Eqs. (109) and (116) in the expression

(t2)α = τ2 (v2)α , (117)

we obtain

(t2)α = det12[gγ δ

]Rαβ (t1)

β . (118)

Now we explain how to calculate (v1)α . From Eqs. (43)

and (100), we write the equations that (v1)α and κ̂1 satisfy:

(κ̂1•1 − κ̂1

)(v1)

1 + κ̂1•2 (v1)2 = 0, (119)

κ̂2•1 (v1)1 +

(κ̂2•2 − κ̂1

)(v1)

2 = 0, (120)(κ̂1•1 − κ̂1

) (κ̂2•2 − κ̂1

)− κ̂2•1κ̂1•2 = 0. (121)

If κ̂1•2 or κ̂1 − κ̂1•1 is nonzero, we can write, from Eq. (119),

[(v1)

α] =

[κ̂1•2

κ̂1 − κ̂1•1

]. (122)

If κ̂1 − κ̂2•2 or κ̂2•1 is nonzero, we can write, from Eq. (120),

[(v1)

α] =

[κ̂1 − κ̂2•2

κ̂2•1

]. (123)

If both κ̂1 − κ̂1•1 and κ̂2•1 are zero, we can write,

[(v1)

α] =

[10

]. (124)

Taking into account the numerical stability issues related todealing with very small numbers, we select one of these threeequations in the order given below.

1. If∣∣κ̂1 − κ̂1•1

∣∣ > ε∣∣κ̂1•2

∣∣, where ε = 10−6, then Eq. (122)is selected.

2. If∣∣κ̂1 − κ̂1•1

∣∣ ≤ ε∣∣κ̂1•2

∣∣ and if∣∣κ̂2•1

∣∣ > ε∣∣κ̂1 − κ̂2•2

∣∣, thenEq. (123) is selected.

3. Otherwise Eq. (124) is selected.

B Derivative of the normal vector

Derivative of the normal vector with respect to ξα can beobtained as follows:

n,α = ∂

∂ξα

(g1 × g2(

g1 × g2) · n

)(125)

= (I − nn) · g1,α × g2 + g1 × g2,α(g1 × g2

) · n (126)

= (I − nn) · g1,α × (n × g1

) + g2,α × (n × g2

)(g1 × g2

) · n(g1 × g2) · n (127)

= (I − nn) · (gβ,α × (n × gβ

))(128)

= (I − nn) · ((gβ,α · gβ)n − (

gβ,α · n) gβ)

(129)

= − (gβ,α · n) gβ + n

(n · gβ

)︸ ︷︷ ︸

=0

(gβ,α · n) (130)

= −gβgβ,α · n (131)

= −gβbαβ. (132)

In the derivation, we used the following relationships, whichgenerally hold:

g1 = g2 × n(g1 × g2

) · n , (133)

g2 = n × g1(g1 × g2

) · n , (134)

(g1 × g2

)· n = ((g1 × g2) · n)−1 . (135)

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